Two stunt drivers drive directly toward each other. At time t=0 the two cars are a distance D apart, car 1 is at rest, and car 2 is moving to the left with speed v0. Car 1 begins to move at t=0, speeding up with a constant acceleration ax. Car 2 continues to move with a constant velocity.

At what time do the two cars collide?

To determine the time at which the two cars collide, we can use the concept of relative motion. Let's break down the problem step by step:

Step 1: Determine the equations of motion for each car.
- For car 1, which starts from rest and undergoes constant acceleration, we can use the equation: x1 = 1/2 * ax * t^2, where x1 represents the distance traveled by car 1 at time t.
- For car 2, which moves at a constant velocity, its equation of motion is simply: x2 = v0 * t, where x2 represents the distance traveled by car 2 at time t.

Step 2: Set up an equation to determine the time of collision.
- The point of collision occurs when the distances traveled by car 1 and car 2 are equal, so we can set up the following equation: 1/2 * ax * t^2 = v0 * t + D, where D represents the initial distance between the two cars.

Step 3: Simplify the equation.
- Multiply both sides of the equation by 2 to eliminate the fraction: ax * t^2 = 2 * v0 * t + 2D.
- Rearrange the equation: ax * t^2 - 2 * v0 * t - 2D = 0.

Step 4: Solve the equation for t using the quadratic formula.
- The equation obtained above is a quadratic equation with respect to t. We can use the quadratic formula: t = (-b ± √(b^2 - 4ac)) / 2a, where a = ax, b = -2 * v0, and c = -2D.

Step 5: Analyze the solutions.
- By solving the quadratic equation, we obtain two possible values for t. We discard any negative solutions since time cannot be negative in this context. The positive solution represents the time at which the two cars collide.

Note: To find a specific numerical solution, you need to know the values of ax, v0, and D. Substituting those values into the equation will give you the time at which the cars collide.